To compare the graphs of the functions \( f(x) = -\sqrt{x} \) and \( g(x) = -\sqrt[3]{x} \), let’s analyze their domains, ranges, and behavior (increasing or decreasing).
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Domains:
- For \( f(x) = -\sqrt{x} \), the domain is \( [0, \infty) \) because the square root is defined for non-negative values.
- For \( g(x) = -\sqrt[3]{x} \), the domain is \( (-\infty, \infty) \) because cube roots are defined for all real numbers.
- Therefore, the graphs do not have the same domains.
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Ranges:
- The range of \( f(x) = -\sqrt{x} \) is \( (-\infty, 0] \) since \( -\sqrt{x} \) takes on all non-positive values as \( x \) varies over its domain.
- The range of \( g(x) = -\sqrt[3]{x} \) is also \( (-\infty, 0] \) because, regardless of whether \( x \) is positive, negative, or zero, \( -\sqrt[3]{x} \) can still take values in this range.
- Therefore, the graphs have the same ranges.
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Behavior (increasing or decreasing):
- The function \( f(x) = -\sqrt{x} \) is a decreasing function because as \( x \) increases, \( -\sqrt{x} \) decreases.
- The function \( g(x) = -\sqrt[3]{x} \) is also a decreasing function for the same reason.
- Therefore, both functions are decreasing.
Based on this analysis, the correct responses are:
- The graphs have the same ranges.
- The graphs are both decreasing.
So, the responses that are true are:
- The graphs have the same ranges.
- The graphs are both decreasing.
You could select the first statement as a response since it matches your options provided.
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